241w-f13_16.1-16.5 (1)

Proof 3 fix a in d set 10 then theorem

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Unformatted text preview: pendent. Proof. (3) ( ). Fix A in D. Set ( ) 10 ∫ . Then Theorem Let be a vector field in an open connected region D in the xy-plane. (1) If F is conservative then (2) If D has no holes and , then F is conservative. 11 Example (1) Let ( ) ( ) (a) Check that is conservative. (b) Find . such that ( (c) Evaluate ∫( ) ) . 12 Example (2) Evaluate ∫ where C is the curve Remark: Read about “conservation of energy”, p. 10611062. 13 16.4 Green’s theorem Theorem Suppose C is a simple closed curve in the plane with positive (counterclockwise) orientation. Assume D is the region bounded by C. Suppose P and Q have continuous 1st order partial derivatives on D. Then ∫ ∬( Proof. 14 ) Example (1) Use Green’s theorem to evaluate ∫( (2) Area of D ∬ ) ∫ 15 (3) Find the area bounded by the ellipse () () (4) Green’s theorem implies that if D has no holes and on D then is conservative. 16 16.6 Curl and Divergence Let 〈 〉 be a vector field in space. Let 〈 (So 〈 〉 〉.) Definition: curl F | | 〈 div F = Example Let 〉 (the divergence of F) 〈 〉. Find div F and curl F. 17 Theorem (1) curl ( ) . (2) If a vector field F is defined on the whole space and curl F = 0 then F is conservative. (3) div (curl F) = 0. Proof. Only (2) is hard to prove. Examples (1) Let F = 〈 conservative. 〉. Note that curl F 18 . So F is not (2) Let F = 〈 conservative. Then find 〉. Show that F is such that 19...
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